Port Hamiltonian Modeling of a Cable Driven Robot

6th IFAC Workshop Lagrangian and Hamiltonian Methods or Nonlinear Control on 6th IFAC on 6th IFAC Workshop Workshop on Lagrangian Lagrangian and and Hamiltonian Hamiltonian Methods Methods or or Nonlinear Control Universidad Técnica Federico Santa María Nonlinear Control on Lagrangian Nonlinear Control 6th IFAC Workshop and Hamiltonian Methods or Available online at www.sciencedirect.com Universidad Técnica Federico Santa María Valparaíso, Chile, May 1-4, 2018 Universidad Técnica Federico Santa Universidad Técnica Federico Santa María María Nonlinear Valparaíso,Control Chile, May 1-4, 2018 Valparaíso, May 1-4, Valparaíso, Chile, Chile, May 1-4, 2018 2018 Universidad Técnica Federico Santa María Valparaíso, Chile, May 1-4, 2018

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IFAC PapersOnLine 51-3 (2018) 161–168

Port Port Port Port

Hamiltonian Modeling of a Hamiltonian Hamiltonian Modeling Modeling of of a a Hamiltonian Modeling of a Cable Driven Robot Cable Cable Driven Driven Robot Robot ∗∗ Driven ChristianCable Schenk ∗ Burak Y¨ uksel Robot Cristian Secchi ∗∗∗ ∗∗∗ Christian Schenk ∗∗ Burak Y¨ uksel ∗∗ ∗∗ Cristian Secchi ∗∗∗

∗ ∗ BurakH. ∗∗ Cristian Christian Y¨ u ksel Secchi Heinrich ulthoff Christian Schenk Schenk BurakH. Y¨ uB¨ ksel Cristian Secchi ∗∗∗ ∗ Heinrich B¨ u lthoff ∗ ∗∗ ∗ ∗ Heinrich H. B¨ u lthoff Christian Schenk Burak Y¨ u ksel Cristian Secchi ∗∗∗ Heinrich H. B¨ ulthoff ∗ ∗ Heinrich H. B¨ u lthoff Max Planck Institute for Biological Cybernetics, Spemanstr. 38, ∗ for Biological Cybernetics, ∗ ∗ Max Max Planck Planck Institute Institute forT¨ Biological Cybernetics, Spemanstr. Spemanstr. 38, 38, 72076, ubingen, Germany. Max Planck Institute for Biological Cybernetics, Spemanstr. 38, 72076, T¨ u ∗ 72076, T¨ ubingen, bingen, Germany. Germany. (e-mail: [email protected], [email protected]) Max Planck Institute for Biological Cybernetics, Spemanstr. 38, 72076, T¨ u bingen, Germany. (e-mail: [email protected]) ∗∗ [email protected], (e-mail: [email protected], [email protected]) GmbH, Zeiloch 20, 76646 Bruchsal, Germany 72076, T¨ u bingen, Germany. (e-mail: [email protected], [email protected]) ∗∗ Volocopter GmbH, Zeiloch 20, 76646 Bruchsal, Germany ∗∗ ∗∗ Volocopter Volocopter GmbH, Zeiloch 20, Germany (previously affiliated with ITK-Engineering GmbH, Germany). (e-mail: [email protected], [email protected]) Volocopter GmbH, Zeiloch 20, 76646 76646 Bruchsal, Bruchsal, Germany (previously affiliated with ITK-Engineering GmbH, Germany). ∗∗ (previously affiliated with ITK-Engineering GmbH, Germany). Germany). (e-mail: [email protected]). Volocopter GmbH, Zeiloch 20, 76646 Bruchsal, Germany (previously affiliated with ITK-Engineering GmbH, (e-mail: [email protected]). ∗∗∗ (e-mail: of [email protected]). Modena and Reggio Emilia, ViaGermany). G. Amendola (previously affiliated with ITK-Engineering GmbH, (e-mail: [email protected]). ∗∗∗ DISMI,University DISMI,University of Modena and Reggio Emilia, Via G. Amendola ∗∗∗ ∗∗∗ of Modena and Reggio Emilia, Via G. 2, DISMI,University 42122, Reggio Emilia, Italy. (e-mail: [email protected]). (e-mail: [email protected]). DISMI,University of Modena and Reggio Emilia, Via G. Amendola Amendola 2, 42122, Reggio Emilia, Italy. (e-mail: [email protected]). ∗∗∗ 2, 42122, Italy. [email protected]). of Modena and Reggio Emilia, Via G. Amendola 2, DISMI,University 42122, Reggio Reggio Emilia, Emilia, Italy. (e-mail: (e-mail: [email protected]). 2, 42122, Reggio Emilia, Italy. (e-mail: [email protected]). Abstract: In this paper we present a generic Port-Hamiltonian (PH) model that includes cable Abstract: In this paper we present aa generic Port-Hamiltonian (PH) model that includes cable Abstract: In this we Port-Hamiltonian (PH) model that dynamics couplings with the platform all cables among cable each Abstract:(in In particular this paper paper elasticity we present presentand a generic generic Port-Hamiltonian (PH)and model that includes includes cable dynamics (in particular elasticity and couplings with the platform and all cables among each dynamics (in particular elasticity and couplings with the the platform and allsimulator. cables among each other) of a(in cable-driven parallel robot (CDPR), which is used as(PH) a motion Moreover Abstract: In this paper we present a generic Port-Hamiltonian model that includes cable dynamics particular elasticity and couplings with platform and all cables among each other) of aa cable-driven parallel robot (CDPR), which is used as aa motion simulator. Moreover other) of cable-driven parallel robot (CDPR), which is used as motion simulator. Moreover we consider changes in the cable robot parameters, i.e. which it’s elasticity, mass and length when the cables dynamics (in particular elasticity and couplings with the platform and all cables among each other) of a cable-driven parallel (CDPR), is used as a motion simulator. Moreover we consider changes in the cable parameters, i.e. it’s elasticity, mass and length when the cables we consider changes in the cable parameters, i.e. it’s elasticity, mass and length when the cables are wound/unwound from the winches. To the best of isour knowledge nobody considered such other) of a cable-driven parallel robot (CDPR), which used as a motion simulator. Moreover we consider changes in the cable parameters, i.e. it’s elasticity, mass and length when the cables are wound/unwound from the winches. To the best of our knowledge nobody considered such aredetailed wound/unwound the winches. To the of knowledge considered such awe and generic model a CDPR PHbest structure motionwhen simulators are consider changes infrom the cable parameters, i.e. it’s elasticity, massSince andnobody length the cables wound/unwound from theof winches. Toin the best of our our before. knowledge nobody considered such aare detailed and generic model of a CDPR in PH structure before. Since motion simulators are a detailed detailed and generic generic model ofwinches. a CDPR CDPR inthe PHproperties, structure before. Sincenobody motion simulators are built to mimic systems with different physical PH modeling can pave the waysuch for are wound/unwound from the To best of our knowledge considered a and model of a in PH structure before. Since motion simulators are built to mimic systems with different physical properties, PH modeling can pave the way for built to mimic systems with different physical properties, PH modeling can pave the way for physics-shaping controllers. a detailed and generic model of a CDPR in PH structure before. Since motion simulators are built to mimic systems with different physical properties, PH modeling can pave the way for physics-shaping controllers. physics-shaping controllers. built to mimic systems with different physical properties, PH modeling can pave the way for physics-shaping controllers. © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. physics-shaping controllers. Keywords: Cable-Driven parallel robots, finite-element modelling, hybrid/switching systems, Keywords: Cable-Driven parallel robots, finite-element modelling, hybrid/switching systems, Keywords: elasticity. Keywords: Cable-Driven Cable-Driven parallel parallel robots, robots, finite-element finite-element modelling, modelling, hybrid/switching hybrid/switching systems, systems, elasticity. elasticity. Keywords: elasticity. Cable-Driven parallel robots, finite-element modelling, hybrid/switching systems, elasticity. 1. INTRODUCTION Although the CableRobot simulator enjoys all the proper1. INTRODUCTION INTRODUCTION Although the the CableRobot CableRobot simulator simulator enjoys enjoys all all the the properproper1. Although ties of a CDPR we listed above, its control is not trivial 1. INTRODUCTION Although the CableRobot simulator enjoys all theaaproperties of aa CDPR CDPR we listed listed above, above, its control control is not not trivial ties of we its is a trivial task. First of all, since thisabove, CDPR carry humans onboard, 1. INTRODUCTION Although the CableRobot simulator enjoys all the properties of a CDPR we listed its control is not a triviala task. First First of of all, all, since since this this CDPR CDPR carry carry humans humans onboard, onboard, task. aaa controller with stability guarantee iscontrol indispensable. Moreties of a CDPR we listed above, its is not a trivial task. First of all, since this CDPR carry humans onboard, controller with stability guarantee is indispensable. Moreguarantee is over aFirst highwith simulation quality requires ahumans high level of Moretrack-a task. of all,stability since this CDPR carry onboard, A Cable-Driven Parallel Robot (CDPR) is a robotic sys- controller controller stability guarantee is indispensable. indispensable. over a high highwith simulation quality requires a high high level level of MoretrackA Cable-Driven Cable-Driven Parallel Parallel Robot Robot (CDPR) (CDPR) is is aa robotic robotic syssys- over a simulation quality requires a of tracking performance for positions, velocities and accelerations A controller with stability guarantee is indispensable. Moretem, which uses elastic cables in a parallel structure over a high simulation quality requires a high level of trackA Cable-Driven Parallel Robot (CDPR) is a robotic sysing performance for positions, velocities and accelerations tem, which which uses uses elastic elastic cables cables in in aa parallel parallel structure structure ing performance for positions, velocities and accelerations spite of parameter uncertainties, unmodeled effects and tem, over a high simulation quality requires a high level of track(instead of rigid serial links) to(CDPR) move a isplatform insysits in ing performance for positions, velocities and accelerations A Cable-Driven Parallel Robot a robotic tem, which uses elastic cables in a parallel structure in spite spite of of parameter parameter uncertainties, uncertainties, unmodeled unmodeled effects effects and (instead of of rigid rigid serial serial links) links) to to move move aa platform platform in in its its in disturbances such as velocities vibrations induced by and the (instead ing performance for positions, and accelerations Cartesian space. These cables are connected at one end to external in spite of parameter uncertainties, unmodeled effects and tem, which uses elastic cables in a parallel structure (instead of rigid serial links) to move a platform in its external disturbances such as vibrations induced by the Cartesian space. These cables are connected at one end to external disturbances such as vibrations induced by the drive train. This makes, first and foremost, an appropriate Cartesian space. These cables are connected at one end to in spite of parameter uncertainties, unmodeled effects and the platform and at the other end to a series of grounded external disturbances such as vibrations induced by the (instead ofspace. rigid serial links) to connected move a platform in its Cartesian These cables are atofone end to drive train. This makes, first and foremost, an appropriate the platform and at the other end to a series grounded drive train. This makes, first and foremost, an appropriate model of the system necessary. the platform and at the other end to a series of grounded external disturbances such as vibrations induced by the winches (consisting of motors and pulleys) and by winding drive train. This makes, first and foremost, an appropriate Cartesian space. These cables are connected at one end to the platform and atofthe otherand endpulleys) to a series model of of the the system system necessary. necessary. winches (consisting motors andofby bygrounded winding model winches (consisting of motors and pulleys) and winding drive train. makes, first and foremost, an appropriate and unwinding the cables the system operator changes Researchers model of theThis system necessary. the platform and at the other end to a series of grounded winches (consisting of motors and pulleys) and by winding adopted several modeling methods, e.g. using and unwinding unwinding the the cables cables the the system system operator operator changes changes Researchers and adopted several modeling modeling methods, methods, e.g. e.g. using using model of theadopted system necessary. the andthe orientation ofand the platform. parallel Researchers winches (consisting ofcables motorsthe pulleys) andThe by winding and position unwinding system operator changes single massless linear several springs modeling (Lamaurymethods, et al. (2013); Diao the position and orientation of the platform. The parallel Researchers adopted several e.g. using the position and orientation of the platform. The parallel single massless linear springs (Lamaury et al. (2013); Diao structure of a CDPR offers some outstanding advantages and unwinding the cables the system operator changes the position and orientation of the platform. The parallel single massless linear springs (Lamaury et Diao and Ma (2009); Weber et al. (2015)),methods, or al. by(2013); considering Researchers adopted several modeling e.g. using structure of a CDPR offers some outstanding advantages single massless linear springs (Lamaury et al. (2013); Diao structure of a CDPR offers some outstanding advantages and Ma (2009); Weber et al. (2015)), or by considering compared to other designs, such as serial manipulators or the position and orientation of the platform. The parallel structure of aother CDPR offers such someasoutstanding advantages and Ma (2009); Weber et al. (2015)), or by considering the mass of the linear cable (Caverly and Forbes (2014); Caverly single massless springs (Lamaury et al. (2013); Diao compared to designs, serial manipulators or and Ma (2009); Weber et al. (2015)), or by considering compared to designs, as serial manipulators or the mass mass of of the the cable cable (Caverly (Caverly and and Forbes Forbes (2014); (2014); Caverly Caverly stewart platforms, including i) they can reach fast accelstructure of CDPR offers such some advantages compared to aother other designs, such asoutstanding serial manipulators or the et al.mass (2015)) and elasticity in three dimensions (Andersen and Ma (2009); Weber et al. (2015)), or by considering stewart platforms, including i) they can reach fast accelthe of the cable (Caverly and Forbes (2014); Caverly stewart including i) can fast accelet al. al. (2015)) (2015)) and and elasticity elasticity in in three three dimensions dimensions (Andersen (Andersen erations, ii)totheir reduced since accompared otherstructure designs, weight such asis manipulators or et stewart platforms, platforms, including i) they they can reach reach fastthe accelal. (2014); Enmark et al. (2011)), or (2014); by taking the theal. mass of the cable (Caverly and Forbes Caverly erations, ii) their their structure weight isserial reduced since the acet (2015)) and elasticity in three dimensions (Andersen erations, ii) structure weight is reduced since the acet al. (2014); Enmark et al. (2011)), or by taking the tuators stand on the ground, iii) they provide modularity stewart platforms, including i) they can reach fastthe accelerations,stand ii) their structure weight is reduced since ac- et al. (2014); Enmark et al. (2011)), or by taking the hysteresis effects and friction into dimensions account (Miermeister al. (2015)) and elasticity in three (Andersen tuators on the ground, iii) they provide modularity et al. (2014); Enmark et al. (2011)), or by taking the tuators stand on the ground, iii) they provide modularity hysteresis effects and friction into account (Miermeister and scalability, iv) they come with great workspace. Varerations, ii) their structure weight is reduced since the actuators stand oniv) the ground, iii) they provide modularity hysteresis effects into account (Miermeister et al. (2014)). At and the friction same time many authors makethe at (2014); Enmark et al. (2011)), or by taking and scalability, they come with great workspace. Varhysteresis effects and friction into account (Miermeister and they come with great workspace. Varet al. al. (2014)). (2014)). At At the the same same time time many many authors authors make make at at ious examples ofiv) CDPRs exploit these properties, such as et tuators stand on the ground, iii) they provide modularity and scalability, scalability, iv) they come with great workspace. Varleast one ofeffects theAt following assumpations that doesn’t give hysteresis and friction into account (Miermeister ious examples of CDPRs exploit these properties, such as et al. (2014)). the same time many authors make at ious examples of CDPRs exploit these properties, such as least one one of of the the following following assumpations assumpations that that doesn’t doesn’t give give the spydercam for sports events (Spydercam (2000)), the and scalability, iv) they come with great workspace. Various examples of CDPRs exploit these properties, such as least consideration toAt the real nature of many the system: et al. (2014)). the same time authors make at the spydercam for sports events (Spydercam (2000)), the least one of the following assumpations that doesn’t give the sports events (2000)), the consideration to to the the real real nature nature of of the the system: system: STRING-MAN for rehabilitation (Surdilovic andsuch Bernious examples offor CDPRs exploit these properties, as consideration the spydercam spydercam for sports events (Spydercam (Spydercam (2000)), the least one of the following assumpations that doesn’t give STRING-MAN for rehabilitation (Surdilovic and Bernconsideration to the real nature of the system: STRING-MAN for rehabilitation (Surdilovic and Bern• Only the planar motion of the CDPR is considered hardt (2004)), for CoGiRo for logistics (El-Ghazaly al. the spydercam events (Spydercam (2000)), the STRING-MAN forsports rehabilitation (Surdilovic and et BernOnly the the to planar motion of of thethe CDPR is considered considered the real nature system: hardt (2004)), CoGiRo CoGiRo for logistics (El-Ghazaly et al. consideration ••• Only planar motion of hardt (2004)), for logistics (El-Ghazaly et al. (see al. (2015)). (2015)) and FAST for telemetry (Duan et al. (2008)). In STRING-MAN for rehabilitation (Surdilovic and BernOnlye.g. theCaverly planar et motion of the the CDPR CDPR is is considered hardt (2004)), CoGiRo for logistics (El-Ghazaly et al. (see e.g. Caverly et al. (2015)). (2015)) and FAST for telemetry (Duan et al. (2008)). In et al. (2015)) and for (Duan al. (2008)). In • (see The elasticity the cables, or the islateral and this paper weFAST are particularly interested inetthe CableRobot Only theCaverly planarof motion of the CDPR considered hardt (2004)), for interested logistics (El-Ghazaly et al. (see e.g. e.g. Caverly al. (2015)). (2015)). (2015)) and for telemetry telemetry (Duan al.CableRobot (2008)). In • The The elasticity ofet the cables, or the the lateral lateral and this paper weFAST areCoGiRo particularly inetthe the • elasticity of the cables, or and this paper we are particularly interested in CableRobot transversal vibrations are negligible (Diao and Ma Simulator (see Fig. 1), which is developed as a motion (see e.g. Caverly et al. (2015)). (2015)) and FAST for telemetry (Duan et al. (2008)). In • transversal The elasticity of the are cables, or the lateral and this paper we areFig. particularly interested in theas CableRobot vibrations negligible (Diao and Ma Simulator (see 1), which is developed a motion vibrations negligible (Diao and Ma Simulator (see which is developed as aa for motion (2009)). In (Schenk et are al. (2016)) the authors show simulator and inFig. use 1), at the Max-Planck-Institute Bio• transversal The elasticity of the cables, or the lateral and this paper we are particularly interested in the CableRobot transversal vibrations are negligible (Diao and Ma Simulator (see Fig. 1), which is developed as motion (2009)). In In (Schenk (Schenk et et al. al. (2016)) (2016)) the the authors authors show show simulator and and in in use use at at the the Max-Planck-Institute Max-Planck-Institute for for BioBio(2009)). simulator that transversal and et lateral vibrations are not neglilogical Cybernetics, T¨ ubingen (Philipp et al. (2016)). This transversal vibrations are negligible (Diao and Ma Simulator (see Fig. 1), which is developed as a motion (2009)). In (Schenk al. (2016)) the authors show simulator and in use at the Max-Planck-Institute for Biothat transversal transversal and and lateral lateral vibrations vibrations are are not not neglineglilogical Cybernetics, Cybernetics, T¨ T¨ ubingen bingen (Philipp (Philipp et et al. al. (2016)). (2016)). This This that logical u gible for In a particular scale of systems, e.g. the one in system simulate the motion of a(Philipp vehicle, et e.g. car or airplane (2009)). (Schenk et al. (2016)) the authors show simulator and in use at the Max-Planck-Institute for Biothat transversal and lateral vibrations are not neglilogical Cybernetics, T¨ u bingen al. (2016)). This gible for for aa particular particular scale scale of of systems, systems, e.g. e.g. the the one one in in system simulate simulate the the motion motion of of aa vehicle, vehicle, e.g. e.g. car car or or airplane airplane gible system Fig. 1; (Masone et al. (2011); Venrooij et al. (2016)) to study the that transversal and lateral vibrations are not neglilogical Cybernetics, T¨ u bingen (Philipp et al. (2016)). This gible for a particular scale of systems, e.g. the one in system simulate the motion of a vehicle, e.g. cartoorstudy airplane Fig. 1; (Masone et al. (2011); Venrooij et al. (2016)) the Fig. 1; (Masone et al. (2011); Venrooij et al. (2016)) to study the human perception ofmotion motion. gible for a particular scale of systems, e.g. the one in system simulate the of a vehicle, e.g. car or airplane Fig. 1; (Masone et al. (2011); Venrooij et al. (2016)) to study the human perception perception of of motion. motion. human Fig. 1; (Masone et al. (2011); Venrooij et al. (2016)) to study the human perception of motion. human perception Copyright © 2018 IFACof motion. 161

Copyright 161 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © © 2018 2018, IFAC IFAC (International Federation of Automatic Control) Copyright © 2018 IFAC 161 Copyright © 2018 IFAC 161 Peer review under responsibility of International Federation of Automatic Control. Copyright © 2018 IFAC 161 10.1016/j.ifacol.2018.06.047

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Fig. 1. CableRobot simulator at Max Planck Institute for Biological Cybernetics, T¨ ubingen (see also Philipp et al. (2016)). • Cables are assumed to be massless. Large scale CDPRs (with steel cables), such as in (Philipp et al. (2016)) and (Hui (2015)), have a very low ratio between platform mass (including mounted equipment) and cable mass. Thus the assumption of massless cables doesn’t hold for large scale systems; • Cable length is constant or varies only in a small range, because the modeling of the cable winding by the winches is a challenging task especially when the elasticity of the cables is taken into account. However in reality as soon as the platform moves in the entire workspace, this assumption doesn’t hold anymore. An interesting recent work in (Godbole et al. (2017)) shows a planar model of a single degree of cable-actuated system based on the Rayleigh-Ritz approach, which accounts for the change in cable masses. The authors take cable vibrations along the cable axis into account but neglect all other variations. In (Caverly et al. (2015)) the authors decouple the cable dynamics from the platform dynamics and assume a high ratio of platform mass and cable mass. This way cable dynamics become negligigble when the platform moves. For systems with a low ratio 1 as in (Cab (2015)) this assumption doesn’t hold anymore. Our goal is to provide a detailed and generic dynamic model of the CDPRs in 3D, which does not make any of the assumptions listed above and contains a very large class of CDPR systems. Furthermore, we put this generic model in PortHamiltonian (PH) form, which reveals system’s energetic and dynamic property in a compact way (van der Schaft and Jeltsema (2014)). To shape the physical properties of that system one can exploit the advantages of a PH representation in the controller synthesis. For example the InterconnectionDamping Assignment-Passivity Based Control (IDA-PBC) method is widely celebrated for this purpose, which does not necessarily need a system in PH form. In (Y¨ uksel et al. (2014)) and (Y¨ uksel (2017)) the authors showed a PH formulation of a quadrotor and used IDA-PBC to reshape its physics for aerial physical interaction tasks. For more about IDA-PBC method we refer the reader to (Ortega et al. (2002)). To the best of our knowledge, this is the first attempt to present the PH model of a CDPR while considering a detailed and generic model of such a system. Here we 1

By low ratio we mean a ratio between total cable mass to platform mass of 3/4

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consider the three dimensional dynamics of the cables using Finite Element Models (FEM) with masses distributed along the cable, based on the lumped mass model from (Schenk et al. (2016)). Furthermore we tackle the challenging task of modeling the cable mass and elasticity changes, by considering a hybrid/switching model of the cables. By presenting a CDPR (especially for motion simulators e.g. the CableRobot Simulator in Fig. 1) in PH form, we also pave the way for the design of physics-shaping controllers, e.g. IDA-PBC. In particular for motion simulators this is a very attractive feature, since a motion simulator mimics the motion of a vehicle e.g. car, airplane or helicopter. Hence we believe a controller capable of shaping the physical properties of such a robotic system into a desired vehicle has a great potential to be acknowledged as a fundamental tool. The paper is structured as in the following. In Section 2 we present a generic model of a CDPR including its kinematics in detail and define the generalized coordinates, reference frames and the system parameters. Then in Section 3 we derive the dynamics of the system and represent it in both Euler-Lagrange (EL) and PH form. To do so, we first compute the kinetic energy of the whole system in Section 3.1, then its potential energy in Section 3.2. Then we formulate both EL and PH models of the system in Section 3.3. We also take the winding process and change of cable mass into account in Section 3.4, which completes the PH model of the CDPR. There we propose a switching/hybrid model of CDPR, whose stability analysis is in the scope of our future works. We finalize our work in Section 4 with useful remarks and future possible directions. 2. CDPR: FRAMES, COORDINATES AND THE KINEMATICS In this section we introduce the CDPR and its kinematic equations. The overall system consists of a platform, carried by m number of cables, that are connected at one end with the platform and at the other end with their motors. Using FEM model we describe the elastic cables as lumped mass spring damper systems with a fixed number of uniformly distributed masses, where for the i-th cable we consider in total ni number of mass elements, with i = {1, 2, · · · , m}. Hence the overall m number of cable mass elements in the system is n = i=1 ni .

Furthermore FW = {OW , xW , yW , zW } represents the world frame, FP = {OP , xP , yP , zP } the frame attached to the geometrical center of the platform (see also left side of Fig. 2) and FC,i = {OC,i , xC,i , yC,i , zC,i } the frame of the i-th cable at the center of the i-th motor, where +y C,i aligns with a vector pointing from the i-th motor to the corresponding attachment point at the platform (see the right side of Fig. 2). Now we introduce the configuration of the platform with W q P ∈ R6 , where left upper superscript 2 implies a representation in FW . Notice that W q P = [W q TP t W q TP r ]T , where W q P t = [xP y P z P ]T ∈ R3 is the position of (the origin of) FP w.r.t. (the origin of) FW (in FW ), and W q P r = [φ θ ψ]T ∈ R3 is the orientation of FP w.r.t. FW (in FW ), in Euler 2 In the rest of the paper the left upper superscript will always imply the reference frame of an element.

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angles. We consider a FEM model for each cable, therefore finite number of individual mass-spring elements 3 . Hence the coordinates of all elements on cable i are C,i q C,i = [C,i q TC,i,1 · · · C,i q TC,i,ni ]T ∈ R3ni , defined in FC,i , where C,i q C,i,j = [C,i q C,i,j,x C,i q C,i,j,y C,i q C,i,j,z ]T ∈ R3 is the 3D translational coordinate 4 of the j-th mass element of the i-th cable 5 . We combine the coordinates of all cables in FW as W q C = [W q TC,1 · · · W q TC,m ]T ∈ R3n and the coordinates of the motors as q M = [θM,1 · · · θM,m ]T ∈ Rm . Hence q = [W q TP W q TC q TM ]T ∈ R6+3n+m represents the generalized coordinates of the system. Considering the rotations, let us first introduce the orientation of FP w.r.t. FW , with the following rotation matrix 6   cψ c θ c ψ sθ sφ − sψ cφ cψ sθ cφ + sψ sφ W R P = sψ cθ sψ sθ sφ + cψ cφ sψ sθ cφ − cψ sφ , (1) −sθ cθ sφ cθ cφ

where c• = cos(•) and s• = sin(•), and W RP ∈ SO(3), and the Euler angels ψ, θ, φ were defined above. Notice that for future computations we also need to define the orientation FC,i w.r.t. FW by two rotation matrices C,i RW,x (θx ) and C,i RW,z (θz ) between individual cable frames and the world frame. Realize that we choose the cable frame in a way that the y-axis aligns with the cable direction. Since we consider the translational motion of the cable elements and neglect rotations around the cable’s yaxis4 , any rotation between cable frames and the world frame will be a function of the rotations around x and z axes. For the sake of simplicity, let us next introduce the following abbreviations: cx = cos(θC,i,x ), cz = cos(θC,i,z ), θx = θC,i,x , θz = θC,i,z where θx represents the rotation of yC,i and yW around xW and θz the rotation of yC,i and yW around zW . Then the rotation matrix defining the orientation of FW in FC,i is C,i

RW (θz , θx ) = C,i RW,x (θx )C,i RW,z (θz ),

where C,i

C,i

RW,x

RW,z

(2)



   1 0 0 100 = 0 cx sx , F xy = 0 1 0 000 0 −sx cx     000 cz s z 0 = −sz cz 0 , F yz = 0 1 0 001 0 0 1

with θx = c1,x (i)π + c2,x (i)cos−1

√

θz = c1,z (i)π + c2,z (i)cos−1

(3)

(F xy C,i RW,z li )T (F xy C,i RW,z li )

√

√

where c• (i) is the i-th element

(C,i RW,z li )T (C,i RW,z li )

(F xy F yz li )T (F xy F yz li )

7

√

(F xy li

)T (F

of c• and

xy li )





(4)

,

3

Of course there are other ways of modeling cables e.g. using linearised distributed parameter models (Starossek (1991); Berlioz and Lamarque (2005); Irvine and Caughey (1974); Yuan et al. (2014)). In this paper we used a FEM model, which was already studied in (Schenk et al. (2016)) 4 In this paper we consider the 3D translational motion of the cables and neglect their rotational ones, e.g. due to the torsions. 5 Analog to C,i q Wq C,i in FC,i of the i-th cable C,i represents the location of all elements on cable i in FW 6 Notice that this rotation matrix can be different depending on the reference frame convention used in the computations. 7 Note that c depends on the configuration of the system. •

163

163

c1,x = 0m ∈ Rm , c2,x = [1 −1 1 −1 1 −1 1 −1] c1,z = [1 1 1 1 0 0 0 0] , c2,z = [−1 −1 1 1 −1 −1 1 1] . The cable vector li (q P ) and the length of each cable can be computed as (5) li (q P ) = (W ai − W q P t − W RP P bi )  ||li (q P )|| = (W ai − W q P t − W RP P bi )T (W ai − W q P t − W RP P bi ), (6) where P bi ∈ R3 and W ai ∈ R3 are the positions of Bi and Ai in FP and FW , respectively. Notice that Bi and Ai are the connection points of the i-th cable to the platform and its motor, respectively (see Fig. 2). Now let us stack the rotation matrices of each cables in one matrix and represent them as in the following C ¯ ˆ W (W q ), · · · , C,m R ˆ W (W q )}, (7) RW (W q P ) = diag{C,1 R P P 





∈R3n×3n

where C,i









m times

ˆ W (W q P ) = diag{C,i RW (W q P ), · · · , R    

∈R3ni ×3ni

C,i

ni times

RW (W q P )}. 

(8)

Notice that above for the i-th cable we consider that C,i,1 RW = . . . = C,i,ni RW = C,i RW , which implies that the vibrations of the individual masses have negligible effects on their orientation. This is a valid assumption for kinematics. Note that we will consider the effect of the vibrations on the system dynamics later in Section 3. Then we can say W ¯ C (W q P )C q C , C q C = C R ¯ W (W q P )W q C , (9) qC = W R and it is true that C ¯ W (W q P )W q˙ C . ¯˙ W (W q P )W q C + C R (10) q˙ C = C R ˙ C¯ W 3n×3n Realize that RW ( q P ) ∈ R in (10) is a matrix with the time derivative of the cable rotation matrices, C,i ˙ RW (W q P ), on the main diagonal. Let us compute this term here for the convenience of the future computations and for the meticulous reader. We have   C,i

C,i ˙ RW,11 (W q P ) ˙ W (W q P ) = C,i R ˙ W,21 (W q P ) R C,i ˙ RW,31 (W q P )

˙ W,12 (W q P ) R ˙ W,22 (W q P ) R C,i ˙ RW,32 (W q )

C,i C,i

P

˙ W,13 (W q P ) R ˙ W,23 (W q P ), R C,i ˙ RW,33 (W q ) C,i

C,i

(11)

P

where for each kl-th entry of this matrix it is T  C,i ∂ RW,kl (W q P ) W C,i ˙ q˙ P RW,kl (W q p ) = ∂ W qP = C,i RW,kl (W q P ) W q˙ P      

∈R6×1 ∈R1×6 ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ with  = [ ∂ W q ] = [ ∂xp ∂yp ∂zp ∂φ Wq ∂θ ∂ψ ] Pt ∂ Pr all elements of C,i RW (W q P ). Hence we write (11) in

following form C,i ˙ RW (W q P ) = D C,i (W q P )S C,i (W q˙ P ) ∈ R3×3 , where  

for the

(12)

C,i RW,11 (W q P ) C,i RW,12 (W q P ) C,i RW,13 (W q P ) D C,i (q p ) = C,i RW,21 (W q P ) C,i RW,22 (W q P ) C,i RW,23 (W q P ) ∈ R3×18 C,i RW,31 (W q P ) C,i RW,32 (W q P ) C,i RW,33 (W q P )

 q˙ P 06×1 06×1 W S C,i (q˙ p ) =  06×1 q˙ P 06×1  ∈ R18×3 . 06×1 06×1 W q˙ P W

Now utilizing the terms S C,i and D C,i computed above in (12); then using S C,i (W q˙ P )

W

   W q˙ P W q C,i,j,x I 6×6 W q C,i,j,x q C,i,j = W q˙ P W q C,i,j,y  = I 6×6 W q C,i,j,y  W q˙ P , W I 6×6 W q C,i,j,z q˙ P W q C,i,j,z       ∈R18×1

Λi (W q C,i )∈R18×6

(13)

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simulator CableRobot Simulator FC,3

Bi

FC,5 mC,i,j

FC,4

FC,6

η C,i,j

A6

lseg,i

zP

FC,2 xP

zW

FP

C,i ¯

qC,i,j

yW

yC,i zC,i

xW FC,1

i -th cable lseg,i

yP B6

FW

lseg,i

C,i η

FC,7

C,i η

η C,i,j =

C,i

FC,8

qC,i,j −

C,i ¯

qC,i,j

xC,i

FC,i A i

winch/motor

Fig. 2. Sketch of the cable robot for m = 8 and the details of an individual cable. On the left, the frame of references for world FW , platform FP and the cables FC,i are shown on a sketch of the CDPR simulator. On the right the i-th cable is represented, whose elastic deflection is taken into account. Hence the true coordinate of an individual (j-th) mass element of the i-th cable in its cable frame is given with C,i qC,i,j . This idea is illustrated by the zoomed sketch placed in the middle of the figure. one can bring (10) into ¯ C (W q )W q˙ + W R( ¯ C (W q )Λ ¯ W q P )W q˙ C q˙ C = D P C P C ¯ C (W q ,W q )W q˙ + Ω ¯ C (W q )W q˙ , ˙ =Γ q˙ C (q, q) P C P P C C

(14)

¯ C (W q ), ¯C = D ¯ C (W q )Λ for all cables where Γ P P ¯C = CR ¯ W (W q P ) and Ω

    W W W ˆ ˆ ¯ D C ( q P ) = diag D C,1 ( q P ), · · · , D C,m ( q P )       m times ∈R3n×18n       W W W ˆ C,i ( q ) = diag D C,i ( q ), · · · , D C,i ( q ) D P P P      ni times

W

(15)

T ˆ T (W q ˆ T (W q ) · · · Λ ¯ (W q )C = [Λ Λ C,m )] C,1 C,m  C   C,1   ∈R18n×6

where TP ∈ R≥0 is the kinetic energy of the platform, TC ∈ R≥0 is the kinetic energy of all cables, and finally TM ∈ R≥0 represents the kinetic energy of the motors. Notice that 1 (17) TP = W q˙ TP W M P (W q P ) W q˙ P , 2 where from (Schenk et al. (2016))

m times

ˆ C (W q ) = [Λ ΛTC,1 (W q C,1 ) · · · Λ TC,1 (W q C,m )]T . Λ C    ni times

with Λ i is available from (13). Notice that in this way, we represent the cable velocities in (14) as sole functions of q ˙ which will become handy for the computations in and q, Section 3. 3. PORT HAMILTONIAN MODELING OF CDPR Since we became familiar with the geometry of the system in Section 2, let us now consider its dynamics as well. In the following, we will first compute the kinetic and potential energies of the overall CDPR system. Since we consider a scleronomous system subject to conservative forces, the sum of its kinetic and potential energies equals its Hamiltonian (Goldstein et al. (2000)). We will use this energetic function for deriving a physically meaningful representation of the system in PH structure.

M P (W q P ) =

We considered that the CDPR consist of a platform, cables and the motors to drive them. Hence the kinetic energy of the overall system is (16) T = T P + TC + TM , 164

mP I 3×3 03×3 03×3 T T (W q P r )W RTP P J P W RP T (W q P r )



(18)

with mP ∈ R+ is the platform mass, I is an identity matrix in proper dimensions, T (W q P r ) is a well-known transformation matrix between angular velocities of the platform and Euler rates (see for instance (Khosravi and Taghirad (2013))), W RP = W RP (W q P r ) is the rotation matrix in (1) and P J P is the rotational inertia of the platform defined in FP . Then the total kinetic energy of the cables is m  1  C,i T C,i TC = q˙ C,i M C,i (W q P )C,i q˙ C,i , 2 i=1

(19)

where C,i M C,i (W q P ) is the inertia matrix of an individual cable in its own coordinate frame, which was previously shown in (Schenk et al. (2016)) as C,i

M C,i (W q P ) = 12 µ diag {lseg,i , 2lseg,i , · · · , 2lseg,i , lseg,i , } ⊗ I 3×3 ,

(20)

where lseg,i will be available from (28) and µ is the mass per length unit of the cable. Notice we wrote the kinetic energy in (20) using the cable coordinates in FC . However recall that in Section 2 we considered the generalized coordinates in FW . Let us then utilize (14) in (19) TC =

3.1 Kinetic Energy



m  1  W T  T Ωi (q) W q˙ C,i q˙ C,i Ω i (q)C,i M c,i (W q P )Ω 2 i=1   Γi (q) W q˙ P + W q˙ TP Γ Ti (q)C,i M C,i (W q P )Γ  Γi (q)W q˙ P , + 2 W q˙ TC,iΩ Ti (q)C,i M C,i (W q P )Γ

where Γ C,i and Ω C,i are available using (15).

(21)

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The kinetic energy of the motors are m 1  ˙2 TM = θ md rd2 2 i=1 M,i

(22)

with the mass md ∈ R+ , the radius of each motor drum rd ∈ R+ . 3.2 Potential Energy The potential energy of the system includes the contribution of the platform Vp ∈ R≥0 and the cables VC ∈ R≥0 including the gravitational and elastic potentials such that the overall potential energy read V (q) = VP (q) + VC (q). (23) Since the motors are fixed to the environment they have no potential energy. The platform potential energy can be computed as VP (q) = mP g q TP [0 0 1 0 0 0], (24) 8 where g ∈ R is the gravity constant . There are two types of potential energies stored in the cables, hence Vc (q) = VCg (q) + VCe (q). The gravitational potential energy of the cables is similar to (24) ni m   VCg (q) = (25) mC,i,j g W q TC,i,j e3 , i=1 j=1

where mC,i,j is the mass of the j-th element of the i-th cable (see Fig. 2) and e3 = [0 0 1]T .

Notice that in this paper we consider the influence of elastic deflections W η C,i = [W η TC,i,1 · · · W η TC,i,ni ]T ∈ R3ni defined in FW and C,iη C,i = [C,iη TC,i,1 · · · C,iη TC,i,ni ]T ∈ R3ni in FC,i (see Fig. 2). Assuming no elastic deflection of ¯ C,i,j in FW pointing from the cable there is a vector W q it’s attachment point at the motor, located at Ai defined in FW to its attachment point at the platform, located at Bi defined in FP . However the true coordinate of the j-th mass element of the i-th cable (w.l.o.g. in FW ) is given with W ¯ C,i,j + W η C,i,j qC,i,j = W q (26) W ¯ C,i + W η C,i , qC,i = W q ¯ C,i = [W q ¯ TC,i,1 · · · W q ¯ TC,i,ni ]T ∈ R3ni and where W q W T W T T η C,i = [ η C,i,1 · · · η C,i,ni ] ∈ R3ni . Now let us show that W η C,i = W η C,i (q), by proofing that it is actually 9 W¯ ¯ C,i (q). First imagine W q ¯ C,i (0) as a vector q C,i = W q containing the initial rest positions of the i-th cable for all mass elements when the cable is fully unwound,   W ai 1 W  (W ai − W ai )  ai − W  ∈ R3ni , (27) ni − 1 ¯ C,i (0) =  q   .   .. W ai  W  T W¯ ¯ C,i,1 (0) ¯ C,i,ni (0) with q C,i (0) = ... Wq q   W ai  = max(W ak − W ai ), k = 1 . . . m, k = i, and W ai ∈ 3 R is the position of Ai in FW . Then, the distance of an individual mass to its neighbor will be   ¯ C,i,k (0) − W q ¯ C,i,k+1 (0) = lseg,i lseg,i,k = W q (28) Wη

8 9

A common approximation on earth is g = 9.81 sm2 ∈ R+ . Notice that W q C,i is already element of q from the definition.

165

165

for k = 1 . . . ni − 1. Since we model the cables with a fixed number of uniformly distributed masses, lseg,i,1 = . . . = lseg,i,k = lseg,i holds. Now we describe the distribution of mass elements by  W q P t + W RP P bi + (ni − 1)lseg,i W d W q P t + W RP P bi + (ni − 2)lseg,i W d     .. W¯ q C,i =  , (29) .   W W P W   q P t + RP bi + lseg,i d W W P q P t + R P bi W

W

W

P

q P t − R P bi i− where W d = |W a is the vector showing ai − W q P t − W R P P b i | the direction of the cable from Bi to Ai , defined in FW . Re¯ C,i (W q P ) = W q ¯ C,i (q). ¯ C,i = W q alize that from (29) it is W q

The overallelastic potential energy stored in the cables m is VCe = i=1 VCe,i , where VCe,i represents the elastic potential energy stored in the i-th cable: 1 C,i ¯ C,i (W q P ))T C,i K η (C,i q C,i − C,i q ¯ C,i (W q P )) ( q C,i − C,i q 2 1 ¯ C,i (W q P ))T W K η (W q C,i − W q ¯ C,i (W q P )), = (W q C,i − W q 2

VCe,i =

(30)

where C,i K η is the stiffness matrix for the i-th cable computed in its own frame, and W K η = C,i RTW (W q P )C,i K η C,i RW (W q P ). (31) Notice that C,i K η was computed in (Schenk et al. (2016)) as C,i K η = C,i K E + C,i K G , where  KE,1 −KE,1  −KE,1 KE,1 + KE,2 −KE,2   C,i .. , KE =  .    −KE,ni −1 KE,ni −1 + KE,ni −KE,ni  −KE,ni KE,ni   KG,1 −KG,1 −KG,1 KG,1 + KG,2 −KG,2    C,i .. , KG =  .    −KG,ni −1 KG,ni −1 + KG,ni −KG,ni  −KG,ni KG,ni 

(32)

(33)

with all the missing entries in the matrix are zeros and   li,j:j+1 lTi,j:j+1 T C,i I 3×3 − KG,i,j = 2 Le,i lseg,i (34) T EA li,j:j+1 li,j:j+1 C,i KE,i,j = , ∀j = 1, . . . , ni − 1, 2 Le,i lseg,i ni  li,j:j+1 2 , li,j:j+1 = W q C,i,j+1 − W q C,i,j , with Le = 1

the Young modulus E of the cable, its nominal 10 crosssectional area A ∈ R+ , and the cable tension T ∈ R.

3.3 EL and PH Modeling of the CDRP Let us show here how one can use the energies from Section 3 in order to obtain both EL and PH modeling of the CDPR system. To do so, we consider the external effects on the system motion, e.g. the control torques of the motors. We define u = [u1 · · · um ]T ∈ Rm as the control input vector, which contains the input torques of the CDPR provided by its motors (see small circles on the left of Fig. 2). Considering the generalized coordinates q, these control inputs enter to the system with control matrix in form of T G = [0m×6 0m×3n Im×m ] ∈ R(6+3n+m)×m . (35) 10 Flattening effects can cause small changes in the length and the cross section of the cable, which are not considered in this paper.

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∂(VP +VCg ) ∂q ∂V − ∂qCe . For

−

where

W

and finally the elastic forces are from E(q) =

the external forces we assume for now f ext = 0, but their involvement to both EL or PH models are trivial. To bring the mechanical system from EL to PH form, we choose a new state x = [q T pT ]T , where p ∈ R6+3n+m is the momentum of the CDPR. Recall ˙ For the meticulous reader let that it is p = M (q)q. ˙ Notice that us reveal the computation of p = M (q)q. T = 12 q˙ T M (q)q˙ = 12 pT M (q)−1 p, where T is available ˙ q) ˙ = p = M (q)q, from (16). Then from the fact that ∂T∂(q, q˙ we can compute that   ¯ P (q) ∗ ∗ M ¯ C (q) ∗  ¯ P C (q) M M (q) = M (37) ¯M 0 0 M = M T (q) ∈ R(6+3n+m)×(6+3n+m) ,

¯ P (q) = W M P (p )+ R6×6  M P m  ΓC,i (q) + Γ TC,i q)C,i M C,i (pP )Γ

W

qC,i,1

W

qC,i,2

qC,i,3

W

qC,i,4

θi

Simulator W W

θi W

W

qC,i,2

qC,i,3

θi qC,i,2

W

W

qC,i,4

Simulator

qC,i,1 W

qC,i,3

W

W

q˙ C,i,1 = [0 − θ˙i rd 0]T

W

qC,i,4

qC,i,1

q˙ C,i,1 =

W

W

q˙ C,i,1 = [0 − θ˙i rd 0]T

Simulator

˙ d 0]T q˙ C,i,2 = W q˙ C,i,3 = [0 − θr

Fig. 3. CDPR is in motion. On the left a sketch of different CDPR configurations are shown. This means that cables are either wound or unwound. For the i-th motor a sketch example of cable winding is shown, where ni = 4. Notice that with further winding more mass elements passes to the motor side, which are assumed to be rigidly attached to the pulley and their contribution to the rest of the cable dynamics are neglected. to inner friction. In R we take inner and outer friction into account. 3.4 Cable Winding In the previous part we presented the dynamics of the CDPR and its PH formalization, without considering the cable winding process.(see Fig. 3). So what happens to the cable dynamics, when the motors wind some of its mass elements? This section is dedicated to suggest a method for dealing cable winding in our model.

i=1

and

¯ C (q) = diag{M ¯ C,1 (q), · · · , M ¯ C,m (q)} R3n×3n  M T C,i ¯ C,i (q) = Ω (q) M C,i (p )Ω R M P Ω C,i (q) i T ¯ T (q)]T ¯ P C (q) = [M ¯ R3n×6  M (q) · · · M 3ni ×3ni

P C,i

for ni = 4

i-th motor CableRobot Simulator

Cable Winding

From the system’s kinetic and potential energies in Sections 3.1 and 3.2 we have the freedom to represent the system in both EL or PH forms. For EL we write ˙ + g(q) + E(q) = Gu + f ext , (36) M (q)¨ q + C(q, q) where we will show later in (37) the generalized inertia matrix M (q), the Coriolis forces ˙ (q)q˙ + ∂T , the gravitational forces are g(q) = ˙ =M C(q, q) ∂q

P C,i

¯ P C,i (q) = Ω T (q)C,i M C,i (p )Γ R3ni ×6  M P Γ C,i (q). C,i Now then let us present the CDPR as a PHS in the following   ∂H(p, q)   q˙  ¯  ∂q = (J − R)  ∂H(p, q)  + G(q, p)u p˙ ∂p (38)   ∂H(p, q)  ∂q ¯ T (q, p)  y=G  ∂H(p, q)  , ∂p where H = T +V ∈ R≥0 is the total energy (for our system ¯ = [0 GT ]T with G also the Hamiltonian) of the system, G is the control input matrix as defined in (35), J = −J T is the skew-symmetric interconnection matrix, and R ∈ R≥0 is the positive semi-definite dissipation matrix. Remark 3.1. There are many reasons for dissipating the energy in a system like ours (see Fig. 1). The most dominant form of dissipation in a mechanical system like a CDPR is friction. On a macroscopic scale friction occurs when two surfaces are in contact and move relative to each other. On a molecular level hysteresis effects in the stress-strain diagram occur when stress is put and released on an object such as a cable. These effects indicate a transformation from kinetic energy to thermal energy due 166

Notice that the masses of the elastic elements disappear 11 and the cable stiffness change during the winding process, as clear from (20). Hence the dynamics for this element will be irrelevant to the motion of the cable in the air (see Fig. 3). For the i-th cable, define a diagonal activation matrix Ai ∈ R(ni ×ni ) , whose elements are either 0 for wounded masses (deactivated ), or 1 (activated ) for free masses. Hence when Ai multiplies with W q C,i the coordinates of the wound mass elements are set to zero. Remark 3.2. (Rigidity assumption:). Notice that the difference between C,i q˙ C,i,j and the motor velocity θ˙i is due to the elasticity of the cables. When the mass element comes very close to the motor, we assume that C,i q˙ C,i,j,y ≈ −θ˙i rd , and in case of cantact with the motor 12 it is C,i ˙ qC,i,j,y = −θ˙i rd . So we assume that a mass element becomes rigid part of the motor when C,i q˙ C,i,j,y = −θ˙i rd , and it decouples from the rest of the cable dynamics. Including Ai , the kinetic energy of the cables in the air becomes 11 What

we mean is that winding changes the mass and the stiffness of the cable hanging in the air, hence the overall system mode. 12 Slipping of the cable on the drum as well as flattening is disregarded.

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1  W T ¯ C,i (q)Ai W q˙ C,i q˙ C,i ATi M 2 i=1   ΓC,i (q)Ai W q˙ P + W q˙ TP ATi Γ TC,i (q)C,i M C,i (W q P )Γ  ¯ P C,i (q)W q˙ P . + 2 W q˙ TC,i ATi M m

TC =

the platform but also the elastic cable dynamics as well. We presented both Euler-Lagrange (EL) and PortHamiltonian (PH) formalism of this model, which — to the best of our knowledge — is done for the first time in the literature on such a detailed level. Especially the PH formalism of a CDPR is a novelty.

(39)

The motor kinetic energy changes as well: m

TM

1  ˙2 = θ (md + 2 i=1 M,i

ni −tr(Ai )



 j=1



mi,j )rd2 ,

(40)



:=0, if tr(Ai )=ni

where mi,j ∈ R+ is the mass of the j-th element of the i-th cable. Then the potential energy of the cables will be m  VCe,i VC (q) = VCg (q) + ni m  

mC,i,j g Ai W q C,i,j .e3

The system dynamics changes with winding/unwinding the cables. We considered this by introducing the concept of activation/deactivation of the cable mass elements. Further improvements for this model, including its stability analysis considering the remarks of Zhao and Hill (2008a,b) is in the scope of our future work. Moreover based on the model presented in this paper, we set the basis for designing an appropriate energy (hence physics) shaping controller — e.g. IDA-PBC — . This could be a possible extension of this paper. REFERENCES

i=1

VCg (q) =

167

(41)

i=1 j=1

1W T η C,i ATi W K η Ai W η C,i , 2 ¯ C,i (W q P ). Notice that for with W η C,i = W q C,i − W q the wound mass elements there are no gravitational and elastic potentials, since i) the motors are fixed in the environment, ii) we neglect slip and elasticity for the cables when C,i q˙ C,i,j,y = −θ˙i rd . The platform kinetic and potential energies are as in (17) and (24), respectively. Remark 3.3. The activation matrix Ai is dynamic and changes when C,i q C,i,j,y = 0 . This is because we choose the y-axis of the cable frame aligned with the cable direction (see right of Fig. 2). Remark 3.4. Since the activation matrix switches discretely (see Remark 3.3) the overall system is a hybrid one. The authors in (Zhao and Hill (2008a,b)) studied the stability and passivity of such systems and it is in the scope of our future works as well. Remark 3.5. Once the entry in the activation matrix Ai changes from 1 to 0 the acceleration of this element equals ¨ C,i,j = 0. This is very reasonable and in line zero, i.e. W q with our assumption. In this case the wound mass elements have no contribution to the cable dynamics. Hence it is C,i ˙ ¨ C,i,j = [0 − θ¨i rd 0]T (see q C,i,j = [0 − θ˙i rd 0]T and C,i q right of Fig. 3). We consider this in the model together with the switching matrix Ai . VCe,i (q) =

Then writing the Hamiltonian as H = T + V = TP + TC + TM + VP + VC , where TP from (17), TC from (39), TM from (40), VP from (24) and VC from (41), we write the system again in form of (38). Note that this model discretely changes for each cable because of the reason explained in Remark 3.3 and we need to study the stability considering Remark 3.4. 4. CONCLUSIONS In this paper we presented a generic model for CableDriven Parallel Robots (CDPR), which includes not only 167

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